Theorem tpid3g 3781

Description: Closed theorem form of tpid3 3782. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
tpid3g	⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3g

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 2814	. 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
2		3mix3 1192	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴))
3	2	a1i 9	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)))
4		abid 2217	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)} ↔ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴))
5	3, 4	imbitrrdi 162	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)}))
6		dftp2 3715	. . . . . 6 ⊢ {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)}
7	6	eleq2i 2296	. . . . 5 ⊢ (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶 ∨ 𝑥 = 𝐷 ∨ 𝑥 = 𝐴)})
8	5, 7	imbitrrdi 162	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝑥 ∈ {𝐶, 𝐷, 𝐴}))
9		eleq1 2292	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
10	8, 9	mpbidi 151	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
11	10	exlimdv 1865	. 2 ⊢ (𝐴 ∈ 𝐵 → (∃𝑥 𝑥 = 𝐴 → 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
12	1, 11	mpd 13	1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ {𝐶, 𝐷, 𝐴})

Syntax hints:   → wi 4   ∨ w3o 1001   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215  {ctp 3668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-tp 3674

This theorem is referenced by:  rngmulrg  13166  srngmulrd  13177  lmodscad  13195  ipsmulrd  13207  ipsipd  13210  topgrptsetd  13227